On the Gibbs properties of Bernoulli convolutions related to $\beta$-numeration in multinacci bases

Abstract

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the $\beta$-numeration. A matrix decomposition of these measures is obtained in the case when $\beta$ is a PV number. We also determine their Gibbs properties for $\beta$ being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

arXiv comment39 pages, 5 figures; to appear in Monatsh. Math